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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.solvers;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.analysis.UnivariateFunction;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.NoBracketingException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.NotStrictlyPositiveException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.NullArgumentException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.25"></a>
<FONT color="green">026</FONT>    <a name="line.26"></a>
<FONT color="green">027</FONT>    /**<a name="line.27"></a>
<FONT color="green">028</FONT>     * Utility routines for {@link UnivariateSolver} objects.<a name="line.28"></a>
<FONT color="green">029</FONT>     *<a name="line.29"></a>
<FONT color="green">030</FONT>     * @version $Id: UnivariateSolverUtils.java 1400850 2012-10-22 11:57:17Z erans $<a name="line.30"></a>
<FONT color="green">031</FONT>     */<a name="line.31"></a>
<FONT color="green">032</FONT>    public class UnivariateSolverUtils {<a name="line.32"></a>
<FONT color="green">033</FONT>        /**<a name="line.33"></a>
<FONT color="green">034</FONT>         * Class contains only static methods.<a name="line.34"></a>
<FONT color="green">035</FONT>         */<a name="line.35"></a>
<FONT color="green">036</FONT>        private UnivariateSolverUtils() {}<a name="line.36"></a>
<FONT color="green">037</FONT>    <a name="line.37"></a>
<FONT color="green">038</FONT>        /**<a name="line.38"></a>
<FONT color="green">039</FONT>         * Convenience method to find a zero of a univariate real function.  A default<a name="line.39"></a>
<FONT color="green">040</FONT>         * solver is used.<a name="line.40"></a>
<FONT color="green">041</FONT>         *<a name="line.41"></a>
<FONT color="green">042</FONT>         * @param function Function.<a name="line.42"></a>
<FONT color="green">043</FONT>         * @param x0 Lower bound for the interval.<a name="line.43"></a>
<FONT color="green">044</FONT>         * @param x1 Upper bound for the interval.<a name="line.44"></a>
<FONT color="green">045</FONT>         * @return a value where the function is zero.<a name="line.45"></a>
<FONT color="green">046</FONT>         * @throws NoBracketingException if the function has the same sign at the<a name="line.46"></a>
<FONT color="green">047</FONT>         * endpoints.<a name="line.47"></a>
<FONT color="green">048</FONT>         * @throws NullArgumentException if {@code function} is {@code null}.<a name="line.48"></a>
<FONT color="green">049</FONT>         */<a name="line.49"></a>
<FONT color="green">050</FONT>        public static double solve(UnivariateFunction function, double x0, double x1)<a name="line.50"></a>
<FONT color="green">051</FONT>            throws NullArgumentException,<a name="line.51"></a>
<FONT color="green">052</FONT>                   NoBracketingException {<a name="line.52"></a>
<FONT color="green">053</FONT>            if (function == null) {<a name="line.53"></a>
<FONT color="green">054</FONT>                throw new NullArgumentException(LocalizedFormats.FUNCTION);<a name="line.54"></a>
<FONT color="green">055</FONT>            }<a name="line.55"></a>
<FONT color="green">056</FONT>            final UnivariateSolver solver = new BrentSolver();<a name="line.56"></a>
<FONT color="green">057</FONT>            return solver.solve(Integer.MAX_VALUE, function, x0, x1);<a name="line.57"></a>
<FONT color="green">058</FONT>        }<a name="line.58"></a>
<FONT color="green">059</FONT>    <a name="line.59"></a>
<FONT color="green">060</FONT>        /**<a name="line.60"></a>
<FONT color="green">061</FONT>         * Convenience method to find a zero of a univariate real function.  A default<a name="line.61"></a>
<FONT color="green">062</FONT>         * solver is used.<a name="line.62"></a>
<FONT color="green">063</FONT>         *<a name="line.63"></a>
<FONT color="green">064</FONT>         * @param function Function.<a name="line.64"></a>
<FONT color="green">065</FONT>         * @param x0 Lower bound for the interval.<a name="line.65"></a>
<FONT color="green">066</FONT>         * @param x1 Upper bound for the interval.<a name="line.66"></a>
<FONT color="green">067</FONT>         * @param absoluteAccuracy Accuracy to be used by the solver.<a name="line.67"></a>
<FONT color="green">068</FONT>         * @return a value where the function is zero.<a name="line.68"></a>
<FONT color="green">069</FONT>         * @throws NoBracketingException if the function has the same sign at the<a name="line.69"></a>
<FONT color="green">070</FONT>         * endpoints.<a name="line.70"></a>
<FONT color="green">071</FONT>         * @throws NullArgumentException if {@code function} is {@code null}.<a name="line.71"></a>
<FONT color="green">072</FONT>         */<a name="line.72"></a>
<FONT color="green">073</FONT>        public static double solve(UnivariateFunction function,<a name="line.73"></a>
<FONT color="green">074</FONT>                                   double x0, double x1,<a name="line.74"></a>
<FONT color="green">075</FONT>                                   double absoluteAccuracy)<a name="line.75"></a>
<FONT color="green">076</FONT>            throws NullArgumentException,<a name="line.76"></a>
<FONT color="green">077</FONT>                   NoBracketingException {<a name="line.77"></a>
<FONT color="green">078</FONT>            if (function == null) {<a name="line.78"></a>
<FONT color="green">079</FONT>                throw new NullArgumentException(LocalizedFormats.FUNCTION);<a name="line.79"></a>
<FONT color="green">080</FONT>            }<a name="line.80"></a>
<FONT color="green">081</FONT>            final UnivariateSolver solver = new BrentSolver(absoluteAccuracy);<a name="line.81"></a>
<FONT color="green">082</FONT>            return solver.solve(Integer.MAX_VALUE, function, x0, x1);<a name="line.82"></a>
<FONT color="green">083</FONT>        }<a name="line.83"></a>
<FONT color="green">084</FONT>    <a name="line.84"></a>
<FONT color="green">085</FONT>        /** Force a root found by a non-bracketing solver to lie on a specified side,<a name="line.85"></a>
<FONT color="green">086</FONT>         * as if the solver was a bracketing one.<a name="line.86"></a>
<FONT color="green">087</FONT>         * @param maxEval maximal number of new evaluations of the function<a name="line.87"></a>
<FONT color="green">088</FONT>         * (evaluations already done for finding the root should have already been subtracted<a name="line.88"></a>
<FONT color="green">089</FONT>         * from this number)<a name="line.89"></a>
<FONT color="green">090</FONT>         * @param f function to solve<a name="line.90"></a>
<FONT color="green">091</FONT>         * @param bracketing bracketing solver to use for shifting the root<a name="line.91"></a>
<FONT color="green">092</FONT>         * @param baseRoot original root found by a previous non-bracketing solver<a name="line.92"></a>
<FONT color="green">093</FONT>         * @param min minimal bound of the search interval<a name="line.93"></a>
<FONT color="green">094</FONT>         * @param max maximal bound of the search interval<a name="line.94"></a>
<FONT color="green">095</FONT>         * @param allowedSolution the kind of solutions that the root-finding algorithm may<a name="line.95"></a>
<FONT color="green">096</FONT>         * accept as solutions.<a name="line.96"></a>
<FONT color="green">097</FONT>         * @return a root approximation, on the specified side of the exact root<a name="line.97"></a>
<FONT color="green">098</FONT>         * @throws NoBracketingException if the function has the same sign at the<a name="line.98"></a>
<FONT color="green">099</FONT>         * endpoints.<a name="line.99"></a>
<FONT color="green">100</FONT>         */<a name="line.100"></a>
<FONT color="green">101</FONT>        public static double forceSide(final int maxEval, final UnivariateFunction f,<a name="line.101"></a>
<FONT color="green">102</FONT>                                       final BracketedUnivariateSolver&lt;UnivariateFunction&gt; bracketing,<a name="line.102"></a>
<FONT color="green">103</FONT>                                       final double baseRoot, final double min, final double max,<a name="line.103"></a>
<FONT color="green">104</FONT>                                       final AllowedSolution allowedSolution)<a name="line.104"></a>
<FONT color="green">105</FONT>            throws NoBracketingException {<a name="line.105"></a>
<FONT color="green">106</FONT>    <a name="line.106"></a>
<FONT color="green">107</FONT>            if (allowedSolution == AllowedSolution.ANY_SIDE) {<a name="line.107"></a>
<FONT color="green">108</FONT>                // no further bracketing required<a name="line.108"></a>
<FONT color="green">109</FONT>                return baseRoot;<a name="line.109"></a>
<FONT color="green">110</FONT>            }<a name="line.110"></a>
<FONT color="green">111</FONT>    <a name="line.111"></a>
<FONT color="green">112</FONT>            // find a very small interval bracketing the root<a name="line.112"></a>
<FONT color="green">113</FONT>            final double step = FastMath.max(bracketing.getAbsoluteAccuracy(),<a name="line.113"></a>
<FONT color="green">114</FONT>                                             FastMath.abs(baseRoot * bracketing.getRelativeAccuracy()));<a name="line.114"></a>
<FONT color="green">115</FONT>            double xLo        = FastMath.max(min, baseRoot - step);<a name="line.115"></a>
<FONT color="green">116</FONT>            double fLo        = f.value(xLo);<a name="line.116"></a>
<FONT color="green">117</FONT>            double xHi        = FastMath.min(max, baseRoot + step);<a name="line.117"></a>
<FONT color="green">118</FONT>            double fHi        = f.value(xHi);<a name="line.118"></a>
<FONT color="green">119</FONT>            int remainingEval = maxEval - 2;<a name="line.119"></a>
<FONT color="green">120</FONT>            while (remainingEval &gt; 0) {<a name="line.120"></a>
<FONT color="green">121</FONT>    <a name="line.121"></a>
<FONT color="green">122</FONT>                if ((fLo &gt;= 0 &amp;&amp; fHi &lt;= 0) || (fLo &lt;= 0 &amp;&amp; fHi &gt;= 0)) {<a name="line.122"></a>
<FONT color="green">123</FONT>                    // compute the root on the selected side<a name="line.123"></a>
<FONT color="green">124</FONT>                    return bracketing.solve(remainingEval, f, xLo, xHi, baseRoot, allowedSolution);<a name="line.124"></a>
<FONT color="green">125</FONT>                }<a name="line.125"></a>
<FONT color="green">126</FONT>    <a name="line.126"></a>
<FONT color="green">127</FONT>                // try increasing the interval<a name="line.127"></a>
<FONT color="green">128</FONT>                boolean changeLo = false;<a name="line.128"></a>
<FONT color="green">129</FONT>                boolean changeHi = false;<a name="line.129"></a>
<FONT color="green">130</FONT>                if (fLo &lt; fHi) {<a name="line.130"></a>
<FONT color="green">131</FONT>                    // increasing function<a name="line.131"></a>
<FONT color="green">132</FONT>                    if (fLo &gt;= 0) {<a name="line.132"></a>
<FONT color="green">133</FONT>                        changeLo = true;<a name="line.133"></a>
<FONT color="green">134</FONT>                    } else {<a name="line.134"></a>
<FONT color="green">135</FONT>                        changeHi = true;<a name="line.135"></a>
<FONT color="green">136</FONT>                    }<a name="line.136"></a>
<FONT color="green">137</FONT>                } else if (fLo &gt; fHi) {<a name="line.137"></a>
<FONT color="green">138</FONT>                    // decreasing function<a name="line.138"></a>
<FONT color="green">139</FONT>                    if (fLo &lt;= 0) {<a name="line.139"></a>
<FONT color="green">140</FONT>                        changeLo = true;<a name="line.140"></a>
<FONT color="green">141</FONT>                    } else {<a name="line.141"></a>
<FONT color="green">142</FONT>                        changeHi = true;<a name="line.142"></a>
<FONT color="green">143</FONT>                    }<a name="line.143"></a>
<FONT color="green">144</FONT>                } else {<a name="line.144"></a>
<FONT color="green">145</FONT>                    // unknown variation<a name="line.145"></a>
<FONT color="green">146</FONT>                    changeLo = true;<a name="line.146"></a>
<FONT color="green">147</FONT>                    changeHi = true;<a name="line.147"></a>
<FONT color="green">148</FONT>                }<a name="line.148"></a>
<FONT color="green">149</FONT>    <a name="line.149"></a>
<FONT color="green">150</FONT>                // update the lower bound<a name="line.150"></a>
<FONT color="green">151</FONT>                if (changeLo) {<a name="line.151"></a>
<FONT color="green">152</FONT>                    xLo = FastMath.max(min, xLo - step);<a name="line.152"></a>
<FONT color="green">153</FONT>                    fLo  = f.value(xLo);<a name="line.153"></a>
<FONT color="green">154</FONT>                    remainingEval--;<a name="line.154"></a>
<FONT color="green">155</FONT>                }<a name="line.155"></a>
<FONT color="green">156</FONT>    <a name="line.156"></a>
<FONT color="green">157</FONT>                // update the higher bound<a name="line.157"></a>
<FONT color="green">158</FONT>                if (changeHi) {<a name="line.158"></a>
<FONT color="green">159</FONT>                    xHi = FastMath.min(max, xHi + step);<a name="line.159"></a>
<FONT color="green">160</FONT>                    fHi  = f.value(xHi);<a name="line.160"></a>
<FONT color="green">161</FONT>                    remainingEval--;<a name="line.161"></a>
<FONT color="green">162</FONT>                }<a name="line.162"></a>
<FONT color="green">163</FONT>    <a name="line.163"></a>
<FONT color="green">164</FONT>            }<a name="line.164"></a>
<FONT color="green">165</FONT>    <a name="line.165"></a>
<FONT color="green">166</FONT>            throw new NoBracketingException(LocalizedFormats.FAILED_BRACKETING,<a name="line.166"></a>
<FONT color="green">167</FONT>                                            xLo, xHi, fLo, fHi,<a name="line.167"></a>
<FONT color="green">168</FONT>                                            maxEval - remainingEval, maxEval, baseRoot,<a name="line.168"></a>
<FONT color="green">169</FONT>                                            min, max);<a name="line.169"></a>
<FONT color="green">170</FONT>    <a name="line.170"></a>
<FONT color="green">171</FONT>        }<a name="line.171"></a>
<FONT color="green">172</FONT>    <a name="line.172"></a>
<FONT color="green">173</FONT>        /**<a name="line.173"></a>
<FONT color="green">174</FONT>         * This method attempts to find two values a and b satisfying &lt;ul&gt;<a name="line.174"></a>
<FONT color="green">175</FONT>         * &lt;li&gt; &lt;code&gt; lowerBound &lt;= a &lt; initial &lt; b &lt;= upperBound&lt;/code&gt; &lt;/li&gt;<a name="line.175"></a>
<FONT color="green">176</FONT>         * &lt;li&gt; &lt;code&gt; f(a) * f(b) &lt; 0 &lt;/code&gt;&lt;/li&gt;<a name="line.176"></a>
<FONT color="green">177</FONT>         * &lt;/ul&gt;<a name="line.177"></a>
<FONT color="green">178</FONT>         * If f is continuous on &lt;code&gt;[a,b],&lt;/code&gt; this means that &lt;code&gt;a&lt;/code&gt;<a name="line.178"></a>
<FONT color="green">179</FONT>         * and &lt;code&gt;b&lt;/code&gt; bracket a root of f.<a name="line.179"></a>
<FONT color="green">180</FONT>         * &lt;p&gt;<a name="line.180"></a>
<FONT color="green">181</FONT>         * The algorithm starts by setting<a name="line.181"></a>
<FONT color="green">182</FONT>         * &lt;code&gt;a := initial -1; b := initial +1,&lt;/code&gt; examines the value of the<a name="line.182"></a>
<FONT color="green">183</FONT>         * function at &lt;code&gt;a&lt;/code&gt; and &lt;code&gt;b&lt;/code&gt; and keeps moving<a name="line.183"></a>
<FONT color="green">184</FONT>         * the endpoints out by one unit each time through a loop that terminates<a name="line.184"></a>
<FONT color="green">185</FONT>         * when one of the following happens: &lt;ul&gt;<a name="line.185"></a>
<FONT color="green">186</FONT>         * &lt;li&gt; &lt;code&gt; f(a) * f(b) &lt; 0 &lt;/code&gt; --  success!&lt;/li&gt;<a name="line.186"></a>
<FONT color="green">187</FONT>         * &lt;li&gt; &lt;code&gt; a = lower &lt;/code&gt; and &lt;code&gt; b = upper&lt;/code&gt;<a name="line.187"></a>
<FONT color="green">188</FONT>         * -- NoBracketingException &lt;/li&gt;<a name="line.188"></a>
<FONT color="green">189</FONT>         * &lt;li&gt; &lt;code&gt; Integer.MAX_VALUE&lt;/code&gt; iterations elapse<a name="line.189"></a>
<FONT color="green">190</FONT>         * -- NoBracketingException &lt;/li&gt;<a name="line.190"></a>
<FONT color="green">191</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.191"></a>
<FONT color="green">192</FONT>         * &lt;p&gt;<a name="line.192"></a>
<FONT color="green">193</FONT>         * &lt;strong&gt;Note: &lt;/strong&gt; this method can take<a name="line.193"></a>
<FONT color="green">194</FONT>         * &lt;code&gt;Integer.MAX_VALUE&lt;/code&gt; iterations to throw a<a name="line.194"></a>
<FONT color="green">195</FONT>         * &lt;code&gt;ConvergenceException.&lt;/code&gt;  Unless you are confident that there<a name="line.195"></a>
<FONT color="green">196</FONT>         * is a root between &lt;code&gt;lowerBound&lt;/code&gt; and &lt;code&gt;upperBound&lt;/code&gt;<a name="line.196"></a>
<FONT color="green">197</FONT>         * near &lt;code&gt;initial,&lt;/code&gt; it is better to use<a name="line.197"></a>
<FONT color="green">198</FONT>         * {@link #bracket(UnivariateFunction, double, double, double, int)},<a name="line.198"></a>
<FONT color="green">199</FONT>         * explicitly specifying the maximum number of iterations.&lt;/p&gt;<a name="line.199"></a>
<FONT color="green">200</FONT>         *<a name="line.200"></a>
<FONT color="green">201</FONT>         * @param function Function.<a name="line.201"></a>
<FONT color="green">202</FONT>         * @param initial Initial midpoint of interval being expanded to<a name="line.202"></a>
<FONT color="green">203</FONT>         * bracket a root.<a name="line.203"></a>
<FONT color="green">204</FONT>         * @param lowerBound Lower bound (a is never lower than this value)<a name="line.204"></a>
<FONT color="green">205</FONT>         * @param upperBound Upper bound (b never is greater than this<a name="line.205"></a>
<FONT color="green">206</FONT>         * value).<a name="line.206"></a>
<FONT color="green">207</FONT>         * @return a two-element array holding a and b.<a name="line.207"></a>
<FONT color="green">208</FONT>         * @throws NoBracketingException if a root cannot be bracketted.<a name="line.208"></a>
<FONT color="green">209</FONT>         * @throws NotStrictlyPositiveException if {@code maximumIterations &lt;= 0}.<a name="line.209"></a>
<FONT color="green">210</FONT>         * @throws NullArgumentException if {@code function} is {@code null}.<a name="line.210"></a>
<FONT color="green">211</FONT>         */<a name="line.211"></a>
<FONT color="green">212</FONT>        public static double[] bracket(UnivariateFunction function,<a name="line.212"></a>
<FONT color="green">213</FONT>                                       double initial,<a name="line.213"></a>
<FONT color="green">214</FONT>                                       double lowerBound, double upperBound)<a name="line.214"></a>
<FONT color="green">215</FONT>            throws NullArgumentException,<a name="line.215"></a>
<FONT color="green">216</FONT>                   NotStrictlyPositiveException,<a name="line.216"></a>
<FONT color="green">217</FONT>                   NoBracketingException {<a name="line.217"></a>
<FONT color="green">218</FONT>            return bracket(function, initial, lowerBound, upperBound, Integer.MAX_VALUE);<a name="line.218"></a>
<FONT color="green">219</FONT>        }<a name="line.219"></a>
<FONT color="green">220</FONT>    <a name="line.220"></a>
<FONT color="green">221</FONT>         /**<a name="line.221"></a>
<FONT color="green">222</FONT>         * This method attempts to find two values a and b satisfying &lt;ul&gt;<a name="line.222"></a>
<FONT color="green">223</FONT>         * &lt;li&gt; &lt;code&gt; lowerBound &lt;= a &lt; initial &lt; b &lt;= upperBound&lt;/code&gt; &lt;/li&gt;<a name="line.223"></a>
<FONT color="green">224</FONT>         * &lt;li&gt; &lt;code&gt; f(a) * f(b) &lt;= 0 &lt;/code&gt; &lt;/li&gt;<a name="line.224"></a>
<FONT color="green">225</FONT>         * &lt;/ul&gt;<a name="line.225"></a>
<FONT color="green">226</FONT>         * If f is continuous on &lt;code&gt;[a,b],&lt;/code&gt; this means that &lt;code&gt;a&lt;/code&gt;<a name="line.226"></a>
<FONT color="green">227</FONT>         * and &lt;code&gt;b&lt;/code&gt; bracket a root of f.<a name="line.227"></a>
<FONT color="green">228</FONT>         * &lt;p&gt;<a name="line.228"></a>
<FONT color="green">229</FONT>         * The algorithm starts by setting<a name="line.229"></a>
<FONT color="green">230</FONT>         * &lt;code&gt;a := initial -1; b := initial +1,&lt;/code&gt; examines the value of the<a name="line.230"></a>
<FONT color="green">231</FONT>         * function at &lt;code&gt;a&lt;/code&gt; and &lt;code&gt;b&lt;/code&gt; and keeps moving<a name="line.231"></a>
<FONT color="green">232</FONT>         * the endpoints out by one unit each time through a loop that terminates<a name="line.232"></a>
<FONT color="green">233</FONT>         * when one of the following happens: &lt;ul&gt;<a name="line.233"></a>
<FONT color="green">234</FONT>         * &lt;li&gt; &lt;code&gt; f(a) * f(b) &lt;= 0 &lt;/code&gt; --  success!&lt;/li&gt;<a name="line.234"></a>
<FONT color="green">235</FONT>         * &lt;li&gt; &lt;code&gt; a = lower &lt;/code&gt; and &lt;code&gt; b = upper&lt;/code&gt;<a name="line.235"></a>
<FONT color="green">236</FONT>         * -- NoBracketingException &lt;/li&gt;<a name="line.236"></a>
<FONT color="green">237</FONT>         * &lt;li&gt; &lt;code&gt; maximumIterations&lt;/code&gt; iterations elapse<a name="line.237"></a>
<FONT color="green">238</FONT>         * -- NoBracketingException &lt;/li&gt;&lt;/ul&gt;&lt;/p&gt;<a name="line.238"></a>
<FONT color="green">239</FONT>         *<a name="line.239"></a>
<FONT color="green">240</FONT>         * @param function Function.<a name="line.240"></a>
<FONT color="green">241</FONT>         * @param initial Initial midpoint of interval being expanded to<a name="line.241"></a>
<FONT color="green">242</FONT>         * bracket a root.<a name="line.242"></a>
<FONT color="green">243</FONT>         * @param lowerBound Lower bound (a is never lower than this value).<a name="line.243"></a>
<FONT color="green">244</FONT>         * @param upperBound Upper bound (b never is greater than this<a name="line.244"></a>
<FONT color="green">245</FONT>         * value).<a name="line.245"></a>
<FONT color="green">246</FONT>         * @param maximumIterations Maximum number of iterations to perform<a name="line.246"></a>
<FONT color="green">247</FONT>         * @return a two element array holding a and b.<a name="line.247"></a>
<FONT color="green">248</FONT>         * @throws NoBracketingException if the algorithm fails to find a and b<a name="line.248"></a>
<FONT color="green">249</FONT>         * satisfying the desired conditions.<a name="line.249"></a>
<FONT color="green">250</FONT>         * @throws NotStrictlyPositiveException if {@code maximumIterations &lt;= 0}.<a name="line.250"></a>
<FONT color="green">251</FONT>         * @throws NullArgumentException if {@code function} is {@code null}.<a name="line.251"></a>
<FONT color="green">252</FONT>         */<a name="line.252"></a>
<FONT color="green">253</FONT>        public static double[] bracket(UnivariateFunction function,<a name="line.253"></a>
<FONT color="green">254</FONT>                                       double initial,<a name="line.254"></a>
<FONT color="green">255</FONT>                                       double lowerBound, double upperBound,<a name="line.255"></a>
<FONT color="green">256</FONT>                                       int maximumIterations)<a name="line.256"></a>
<FONT color="green">257</FONT>            throws NullArgumentException,<a name="line.257"></a>
<FONT color="green">258</FONT>                   NotStrictlyPositiveException,<a name="line.258"></a>
<FONT color="green">259</FONT>                   NoBracketingException {<a name="line.259"></a>
<FONT color="green">260</FONT>            if (function == null) {<a name="line.260"></a>
<FONT color="green">261</FONT>                throw new NullArgumentException(LocalizedFormats.FUNCTION);<a name="line.261"></a>
<FONT color="green">262</FONT>            }<a name="line.262"></a>
<FONT color="green">263</FONT>            if (maximumIterations &lt;= 0)  {<a name="line.263"></a>
<FONT color="green">264</FONT>                throw new NotStrictlyPositiveException(LocalizedFormats.INVALID_MAX_ITERATIONS, maximumIterations);<a name="line.264"></a>
<FONT color="green">265</FONT>            }<a name="line.265"></a>
<FONT color="green">266</FONT>            verifySequence(lowerBound, initial, upperBound);<a name="line.266"></a>
<FONT color="green">267</FONT>    <a name="line.267"></a>
<FONT color="green">268</FONT>            double a = initial;<a name="line.268"></a>
<FONT color="green">269</FONT>            double b = initial;<a name="line.269"></a>
<FONT color="green">270</FONT>            double fa;<a name="line.270"></a>
<FONT color="green">271</FONT>            double fb;<a name="line.271"></a>
<FONT color="green">272</FONT>            int numIterations = 0;<a name="line.272"></a>
<FONT color="green">273</FONT>    <a name="line.273"></a>
<FONT color="green">274</FONT>            do {<a name="line.274"></a>
<FONT color="green">275</FONT>                a = FastMath.max(a - 1.0, lowerBound);<a name="line.275"></a>
<FONT color="green">276</FONT>                b = FastMath.min(b + 1.0, upperBound);<a name="line.276"></a>
<FONT color="green">277</FONT>                fa = function.value(a);<a name="line.277"></a>
<FONT color="green">278</FONT>    <a name="line.278"></a>
<FONT color="green">279</FONT>                fb = function.value(b);<a name="line.279"></a>
<FONT color="green">280</FONT>                ++numIterations;<a name="line.280"></a>
<FONT color="green">281</FONT>            } while ((fa * fb &gt; 0.0) &amp;&amp; (numIterations &lt; maximumIterations) &amp;&amp;<a name="line.281"></a>
<FONT color="green">282</FONT>                    ((a &gt; lowerBound) || (b &lt; upperBound)));<a name="line.282"></a>
<FONT color="green">283</FONT>    <a name="line.283"></a>
<FONT color="green">284</FONT>            if (fa * fb &gt; 0.0) {<a name="line.284"></a>
<FONT color="green">285</FONT>                throw new NoBracketingException(LocalizedFormats.FAILED_BRACKETING,<a name="line.285"></a>
<FONT color="green">286</FONT>                                                a, b, fa, fb,<a name="line.286"></a>
<FONT color="green">287</FONT>                                                numIterations, maximumIterations, initial,<a name="line.287"></a>
<FONT color="green">288</FONT>                                                lowerBound, upperBound);<a name="line.288"></a>
<FONT color="green">289</FONT>            }<a name="line.289"></a>
<FONT color="green">290</FONT>    <a name="line.290"></a>
<FONT color="green">291</FONT>            return new double[] {a, b};<a name="line.291"></a>
<FONT color="green">292</FONT>        }<a name="line.292"></a>
<FONT color="green">293</FONT>    <a name="line.293"></a>
<FONT color="green">294</FONT>        /**<a name="line.294"></a>
<FONT color="green">295</FONT>         * Compute the midpoint of two values.<a name="line.295"></a>
<FONT color="green">296</FONT>         *<a name="line.296"></a>
<FONT color="green">297</FONT>         * @param a first value.<a name="line.297"></a>
<FONT color="green">298</FONT>         * @param b second value.<a name="line.298"></a>
<FONT color="green">299</FONT>         * @return the midpoint.<a name="line.299"></a>
<FONT color="green">300</FONT>         */<a name="line.300"></a>
<FONT color="green">301</FONT>        public static double midpoint(double a, double b) {<a name="line.301"></a>
<FONT color="green">302</FONT>            return (a + b) * 0.5;<a name="line.302"></a>
<FONT color="green">303</FONT>        }<a name="line.303"></a>
<FONT color="green">304</FONT>    <a name="line.304"></a>
<FONT color="green">305</FONT>        /**<a name="line.305"></a>
<FONT color="green">306</FONT>         * Check whether the interval bounds bracket a root. That is, if the<a name="line.306"></a>
<FONT color="green">307</FONT>         * values at the endpoints are not equal to zero, then the function takes<a name="line.307"></a>
<FONT color="green">308</FONT>         * opposite signs at the endpoints.<a name="line.308"></a>
<FONT color="green">309</FONT>         *<a name="line.309"></a>
<FONT color="green">310</FONT>         * @param function Function.<a name="line.310"></a>
<FONT color="green">311</FONT>         * @param lower Lower endpoint.<a name="line.311"></a>
<FONT color="green">312</FONT>         * @param upper Upper endpoint.<a name="line.312"></a>
<FONT color="green">313</FONT>         * @return {@code true} if the function values have opposite signs at the<a name="line.313"></a>
<FONT color="green">314</FONT>         * given points.<a name="line.314"></a>
<FONT color="green">315</FONT>         * @throws NullArgumentException if {@code function} is {@code null}.<a name="line.315"></a>
<FONT color="green">316</FONT>         */<a name="line.316"></a>
<FONT color="green">317</FONT>        public static boolean isBracketing(UnivariateFunction function,<a name="line.317"></a>
<FONT color="green">318</FONT>                                           final double lower,<a name="line.318"></a>
<FONT color="green">319</FONT>                                           final double upper)<a name="line.319"></a>
<FONT color="green">320</FONT>            throws NullArgumentException {<a name="line.320"></a>
<FONT color="green">321</FONT>            if (function == null) {<a name="line.321"></a>
<FONT color="green">322</FONT>                throw new NullArgumentException(LocalizedFormats.FUNCTION);<a name="line.322"></a>
<FONT color="green">323</FONT>            }<a name="line.323"></a>
<FONT color="green">324</FONT>            final double fLo = function.value(lower);<a name="line.324"></a>
<FONT color="green">325</FONT>            final double fHi = function.value(upper);<a name="line.325"></a>
<FONT color="green">326</FONT>            return (fLo &gt;= 0 &amp;&amp; fHi &lt;= 0) || (fLo &lt;= 0 &amp;&amp; fHi &gt;= 0);<a name="line.326"></a>
<FONT color="green">327</FONT>        }<a name="line.327"></a>
<FONT color="green">328</FONT>    <a name="line.328"></a>
<FONT color="green">329</FONT>        /**<a name="line.329"></a>
<FONT color="green">330</FONT>         * Check whether the arguments form a (strictly) increasing sequence.<a name="line.330"></a>
<FONT color="green">331</FONT>         *<a name="line.331"></a>
<FONT color="green">332</FONT>         * @param start First number.<a name="line.332"></a>
<FONT color="green">333</FONT>         * @param mid Second number.<a name="line.333"></a>
<FONT color="green">334</FONT>         * @param end Third number.<a name="line.334"></a>
<FONT color="green">335</FONT>         * @return {@code true} if the arguments form an increasing sequence.<a name="line.335"></a>
<FONT color="green">336</FONT>         */<a name="line.336"></a>
<FONT color="green">337</FONT>        public static boolean isSequence(final double start,<a name="line.337"></a>
<FONT color="green">338</FONT>                                         final double mid,<a name="line.338"></a>
<FONT color="green">339</FONT>                                         final double end) {<a name="line.339"></a>
<FONT color="green">340</FONT>            return (start &lt; mid) &amp;&amp; (mid &lt; end);<a name="line.340"></a>
<FONT color="green">341</FONT>        }<a name="line.341"></a>
<FONT color="green">342</FONT>    <a name="line.342"></a>
<FONT color="green">343</FONT>        /**<a name="line.343"></a>
<FONT color="green">344</FONT>         * Check that the endpoints specify an interval.<a name="line.344"></a>
<FONT color="green">345</FONT>         *<a name="line.345"></a>
<FONT color="green">346</FONT>         * @param lower Lower endpoint.<a name="line.346"></a>
<FONT color="green">347</FONT>         * @param upper Upper endpoint.<a name="line.347"></a>
<FONT color="green">348</FONT>         * @throws NumberIsTooLargeException if {@code lower &gt;= upper}.<a name="line.348"></a>
<FONT color="green">349</FONT>         */<a name="line.349"></a>
<FONT color="green">350</FONT>        public static void verifyInterval(final double lower,<a name="line.350"></a>
<FONT color="green">351</FONT>                                          final double upper)<a name="line.351"></a>
<FONT color="green">352</FONT>            throws NumberIsTooLargeException {<a name="line.352"></a>
<FONT color="green">353</FONT>            if (lower &gt;= upper) {<a name="line.353"></a>
<FONT color="green">354</FONT>                throw new NumberIsTooLargeException(LocalizedFormats.ENDPOINTS_NOT_AN_INTERVAL,<a name="line.354"></a>
<FONT color="green">355</FONT>                                                    lower, upper, false);<a name="line.355"></a>
<FONT color="green">356</FONT>            }<a name="line.356"></a>
<FONT color="green">357</FONT>        }<a name="line.357"></a>
<FONT color="green">358</FONT>    <a name="line.358"></a>
<FONT color="green">359</FONT>        /**<a name="line.359"></a>
<FONT color="green">360</FONT>         * Check that {@code lower &lt; initial &lt; upper}.<a name="line.360"></a>
<FONT color="green">361</FONT>         *<a name="line.361"></a>
<FONT color="green">362</FONT>         * @param lower Lower endpoint.<a name="line.362"></a>
<FONT color="green">363</FONT>         * @param initial Initial value.<a name="line.363"></a>
<FONT color="green">364</FONT>         * @param upper Upper endpoint.<a name="line.364"></a>
<FONT color="green">365</FONT>         * @throws NumberIsTooLargeException if {@code lower &gt;= initial} or<a name="line.365"></a>
<FONT color="green">366</FONT>         * {@code initial &gt;= upper}.<a name="line.366"></a>
<FONT color="green">367</FONT>         */<a name="line.367"></a>
<FONT color="green">368</FONT>        public static void verifySequence(final double lower,<a name="line.368"></a>
<FONT color="green">369</FONT>                                          final double initial,<a name="line.369"></a>
<FONT color="green">370</FONT>                                          final double upper)<a name="line.370"></a>
<FONT color="green">371</FONT>            throws NumberIsTooLargeException {<a name="line.371"></a>
<FONT color="green">372</FONT>            verifyInterval(lower, initial);<a name="line.372"></a>
<FONT color="green">373</FONT>            verifyInterval(initial, upper);<a name="line.373"></a>
<FONT color="green">374</FONT>        }<a name="line.374"></a>
<FONT color="green">375</FONT>    <a name="line.375"></a>
<FONT color="green">376</FONT>        /**<a name="line.376"></a>
<FONT color="green">377</FONT>         * Check that the endpoints specify an interval and the end points<a name="line.377"></a>
<FONT color="green">378</FONT>         * bracket a root.<a name="line.378"></a>
<FONT color="green">379</FONT>         *<a name="line.379"></a>
<FONT color="green">380</FONT>         * @param function Function.<a name="line.380"></a>
<FONT color="green">381</FONT>         * @param lower Lower endpoint.<a name="line.381"></a>
<FONT color="green">382</FONT>         * @param upper Upper endpoint.<a name="line.382"></a>
<FONT color="green">383</FONT>         * @throws NoBracketingException if the function has the same sign at the<a name="line.383"></a>
<FONT color="green">384</FONT>         * endpoints.<a name="line.384"></a>
<FONT color="green">385</FONT>         * @throws NullArgumentException if {@code function} is {@code null}.<a name="line.385"></a>
<FONT color="green">386</FONT>         */<a name="line.386"></a>
<FONT color="green">387</FONT>        public static void verifyBracketing(UnivariateFunction function,<a name="line.387"></a>
<FONT color="green">388</FONT>                                            final double lower,<a name="line.388"></a>
<FONT color="green">389</FONT>                                            final double upper)<a name="line.389"></a>
<FONT color="green">390</FONT>            throws NullArgumentException,<a name="line.390"></a>
<FONT color="green">391</FONT>                   NoBracketingException {<a name="line.391"></a>
<FONT color="green">392</FONT>            if (function == null) {<a name="line.392"></a>
<FONT color="green">393</FONT>                throw new NullArgumentException(LocalizedFormats.FUNCTION);<a name="line.393"></a>
<FONT color="green">394</FONT>            }<a name="line.394"></a>
<FONT color="green">395</FONT>            verifyInterval(lower, upper);<a name="line.395"></a>
<FONT color="green">396</FONT>            if (!isBracketing(function, lower, upper)) {<a name="line.396"></a>
<FONT color="green">397</FONT>                throw new NoBracketingException(lower, upper,<a name="line.397"></a>
<FONT color="green">398</FONT>                                                function.value(lower),<a name="line.398"></a>
<FONT color="green">399</FONT>                                                function.value(upper));<a name="line.399"></a>
<FONT color="green">400</FONT>            }<a name="line.400"></a>
<FONT color="green">401</FONT>        }<a name="line.401"></a>
<FONT color="green">402</FONT>    }<a name="line.402"></a>




























































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